The msparse_series_sol command returns a set of m-sparse power series solutions of the given linear ordinary differential equation with polynomial coefficients.

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If ode is an expression, then it is equated to zero.

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The routine returns an error message if the differential equation ode does not satisfy the following conditions.

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ode must be homogeneous and linear in var

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ode must have polynomial coefficients in the independent variable of var, for example, x

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The coefficients of ode must be either rational numbers or depend rationally on one or more parameters.

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A homogeneous linear ordinary differential equation with coefficients that are polynomials in x has a linear space of formal power series solutions ∑n&equals;0∞&ApplyFunction;v&ApplyFunction;n&InvisibleTimes;Pn&ApplyFunction;x where Pn&ApplyFunction;x is one of x−an, x−ann&excl;, 1xn, or 1xn&InvisibleTimes;n&excl;, a is the expansion point, and the sequence v&ApplyFunction;n satisfies a homogeneous linear recurrence.

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This routine selects such formal power series solutions where for an integer m≥2 there is an integer i such that

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v&ApplyFunction;n≠0 only if n−imodm&equals;0, and

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v&ApplyFunction;n&plus;1&InvisibleTimes;m&plus;i&equals;p&ApplyFunction;n&InvisibleTimes;v&ApplyFunction;m&InvisibleTimes;n&plus;i for all sufficiently large n, where p&ApplyFunction;n is a rational function.

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The m-sparse power series is represented by an FPSstruct data-structure (see Slode[FPseries]):

v&ApplyFunction;0,...,v&ApplyFunction;M are expressions, the initial series coefficients,

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M is a nonnegative integer, and

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s is an integer such that M&plus;1≤m&InvisibleTimes;s&plus;N.

Options

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x=a or 'point'=a

Specifies the expansion point a. The default is a&equals;0. It can be an algebraic number, depending rationally on some parameters, or ∞.

If this option is given, then the command returns a set of m-sparse power series solutions at the given point a. Otherwise, it returns a set of m-sparse power series solutions for all possible points that are determined by Slode[candidate_mpoints](ode,var).

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'sparseorder'=m0

Specifies an integer m0. If this option is given, then the procedure computes a set of m-sparse power series solutions with m&equals;m0 only. Otherwise, it returns a set of m-sparse power series solution for all possible values of m.

If both an expansion point and a sparse order are given, then the command can also compute a set of m-sparse series solutions for an inhomogeneous equation with polynomial coefficients and a right-hand side that is rational in the independent variable x. Otherwise, the equation has to be homogeneous.

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'free'=C

Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name _C. Note that the number of free variables may be less than the order of the given equation if the expansion point is singular.